**1. Introduction**

The stagnation point flow has attracted vast attention from many researchers because of its broad applications in both industrial and scientific applications. Some of the real-world applications of the stagnation point flow lie in the polymer industry, extrusion processes, plane counter-jets, and numerous forms of hydrodynamic modelling in engineering uses ([1–3]). An exact solution of the steady two-dimensional stagnation-point flow towards a solid surface in moving fluid was first discovered by Hiemenz [4] in 1911. In his particular study, the Navier–Stokes equations are reduced to non-linear ordinary di fferential equations by using a similarity transformation. The remarkable work done by Hiemenz [4] was extended by Homann [5], who started the classical work of three-dimensional stagnation point flow for the axisymmetric case. Meanwhile, the flow in the neighbourhood of a particular stagnation point on a surface was explored by Howarth [6], focusing on the non-axisymmetric

three-dimensional flow near the stagnation region. Fast forward to 1961, the work of Howarth [6] was criticized by Davey [7], who indicated a mistake in Howarth's paper exposing that the results in the region −1 ≤ *c* ≤ 0 are unable to be achieved from those discovered for 0 ≤ *c* ≤ 1 as reported in the study. In conjunction with these findings, Davey and Schofield [8] initiated the study of these saddle point solutions and justified the existence of the non-uniqueness solution. In another study, Weidman [9] modified the Homann's axisymmetric outer potential stagnation-point flow for non-axisymmetric stagnation flow of the strain rate. The study revealed a new clan of asymmetric viscous stagnation point flows liable on the shear rate ratio, γ = *b*/*a* where −∞ ≤ γ ≤ <sup>∞</sup>, *a* is the strain rate and *b* is the shear rate. An analysis of unsteady heat transmission in non-axisymmetric Homann stagnation-point flows of a viscous fluid over a rigid plate was investigated by Mahapatra and Sidui [10], and recently, an investigation on the non-axisymmetric Homann stagnation-point flows of a viscoelastic fluid towards a fixed plate was conducted by Mahapatra and Sidui [11].

Ever since the evolution study of the stagnation point flow with the presence of dual solutions by Davey [7], various works concerning the stagnation point flow towards a shrinking sheet were introduced. Wang [12] considered two-dimensional stagnation point flow on a two-dimensional shrinking sheet and axisymmetric stagnation point flow on an axisymmetric shrinking sheet, while Mahapatra and Sidui [13] assessed unsteady heat transfer in non-axisymmetric Homann stagnation-point flow towards a stretching/shrinking sheet with stability analysis. The continuous effort was carried out by Khashi'ie et al. [14] who examined the three-dimensional non-axisymmetric Homann stagnation point flow and heat transfer past a stretching/shrinking sheet using hybrid nanofluid. Meanwhile, Zaimi and Ishak [15] scrutinized the slip e ffects on the stagnation point flow towards a stretching vertical sheet. Nevertheless, explorations on the stagnation point flow keep evolving in various ways and have been working still because of its importance in massive engineering applications and also in the magnetohydrodynamics (MHD) flow field. A comprehensive study of the literature on the related works was reviewed by [16–19].

A fluid that is heated by electric energy in the occurrence of a vigorous magnetic field, such as crystal growth in melting, is essential in the industrial sector. The interaction of electrical currents and magnetic fields generates the divergence of Lorentz forces during the movement of fluid. In accordance with this phenomenon, MHD describes the hydrodynamics of a conducting fluid in the presence of a magnetic field. The examinations of MHD flow are very significant due to its massive number of uses implicating the magnetic e ffect in industrial and engineering areas, such as MHD electricity generators, sterilization tools, magnetic resonance graphs, MHD flow meters, and also in granular insulation (see [20,21]). The goods of the end product depend immensely on the rate of cooling involved in these processes, managed by the application of the magnetic field and electrically conducting fluids. The study of MHD flow in the Newtonian fluid was first carried out by Pavlov [22], who investigated the magnetohydrodynamic flow of an impressible viscous fluid caused by deformation of a surface. Chakrabarti and Gupta [23] broadened the study of hydromagnetic flow and heat transfer over a stretching sheet, followed by Vajravelu [24] who widened the hydromagnetic flow study over a continuous, moving, porous flat surface. Andersson, in 1995, introduced an exact solution of the Navier-Stokes equations for magnetohydrodynamic flow [25], and Lok et al. [26] analyzed the MHD stagnation-point flow towards a shrinking sheet using the Keller-box method and proved the existence of multiple (dual) solutions for small values of the magnetic field parameter for the shrinking case. Recently, Almutairi et al. [27] studied the influence of second-order velocity slip on the MHD flow of a nanofluid in a porous medium by considering the homogeneous-heterogeneous reactions. On the other hand, the impact of nonlinear and temperature jump on non-Newtonian MHD nanofluid flow and heat transfer past a stretched thin sheet was examined by Zhu et al. [28]

Over the last few decades, the researcher has experienced tremendous scholarly devotion to the study of heat transfer fluid. Recent demand for a high-e fficiency refrigeration system and the ine ffectiveness of traditional thermal conduction fluids encouraged analysts to discover another heat transfer fluid. Choi and Eastman [29] launched the exploration of nanofluids and illustrated the

presence of suspended nanoparticles in a carrier fluid. This pioneering study led to the verdict of the colloidal suspension of intensely small-sized particles, for instance, carbon nanotubes, metals, oxides, and carbides, into the based fluid, which may ensure access to an advanced course of nanotechnology-based heat transfer media (see [30,31]). The eccentric features of nanofluids have gained grea<sup>t</sup> acknowledgement in various engineering, medical, and industrial applications like engine cooling, diesel generator e fficiency, micro-manufacturing, solar water heating, cancer treatment, nuclear reactors, and diverse types of heat exchangers ([32–34]). Due to the massive potential for the applications of nanofluids, Choi and Eastmen [29] developed a mathematical model of nanofluids, which allowed Buongiorno [31] to contribute to heat transfer analysis in nanofluids by introducing the non-homogeneous model for transport and heat transfer phenomena in nanofluids with turbulence applications.

Recently, an expansion of new engineered nanofluids was achieved by dispersing composite nanopowder or dissimilar nanoparticles with sizes between 1 and 10nm in the base fluid [29]; it is known as a hybrid nanofluid. The hybrid nanofluid is a modern technology fluid that may o ffer better heat transfer performance and thermal physical properties. The progress related to the preparation methods of hybrid nanofluids, thermo-physical properties of hybrid nanofluids, and current applications of hybrid nanofluids was published by Sarkar et al. [35] and Sidik et al. [36]. In another study, Huminic and Huminic [37] highlighted the essential applications of hybrid nanofluids, such as in heat pipes, mini-channel heat sinks, plate heat exchangers, air conditioning systems, tubular heat exchangers, shell and tube heat exchangers, tube in a tube heat exchangers, and coiled heat exchangers. Turcu et al. [38], for the first time testified the hybrid nanocomposite particle synthesis, which consisted of two di fferent hybrids, polypyrrole-carbon nanotube (PPY-CNT) nanocomposite and multi-walled carbon nanotube (MWCNT) on magnetic Fe3O4 nanoparticles. In the following year, Yen et al. [39] inspected the effect of hybrid nanofluids in channel flow numerically. Devi and Devi [40] analysed the problem of hydromagnetic hybrid nanofluid (Cu-Al2O3/water) flow on a permeable stretching sheet subject to Newtonian heating, and they continued the investigation to improve the heat transfer in hybrid nanofluid flow past a stretching sheet [41]. Subsequently, Yousefi et al. [42] reviewed on the stagnation point flow of an aqueous titania-copper hybrid nanofluid toward a wavy cylinder. At the same time, Khashi'ie et al. [43] performed a numerical study on the heat transfer and boundary layer flow of axisymmetric hybrid nanofluids driven by a stretching/shrinking disc. A detailed documentary on the numerical study of hybrid nanofluid flow and heat transfer is reviewed by [44–48].

Many practical situations, such as a sudden stretching of the plate or temperature change of the plate, involved unsteady conditions of the heat transfer flow. Cai et al. [49] explained that the flow in the viscous boundary layer near the plate would slowly be enlarged if the surface was extended unexpectedly, and hence converted into a steady flow after a certain interval. Technically, we believe that the consideration of physical quantities related to time is crucial in mathematical modeling and analysis, which is acknowledged in the formulation of this research problem.

Motivated by the work by Mahapatra and Sidui [13], this study aims to inspect the unsteady MHD non-axisymmetric Homann stagnation point of a hybrid nanofluid in three-dimensional flow. The proposed hybrid nanofluid model is adapted from Devi and Devi [40] and Hayat and Nadeem [50], recognized by suspending varied nanoparticles, namely alumina (Al2O3) and copper (Cu), in the base fluid (water). To the best of the authors' knowledge, no attempt has been made to examine the heat transfer and fluid flow of the hybrid nanofluid (Al2O3-Cu/H2O) considering the unsteady parameter in non-axisymmetric Homann stagnation point flow. This possibly will benefit future works on choosing a significant parameter to enhance the heat transfer performance in the modern industry. The novelty of this study can also be seen in the discovery of dual solutions and the execution of stability analysis. Ultimately, this research is highly claimed to be authentic and original.
